Commuting idempotents, square-free modules, and the exchange property

We give a criterion for when idempotents of a ring R   which commute modulo the Jacobson radical J(R)J(R) can be lifted to commuting idempotents of R  . If such lifting is possible, we give extra information about the lifts. A “half-commuting” analogue is also proven, and this is used to give sufficient conditions for a ring to have the internal exchange property. In particular, we show that if R/J(R)R/J(R) is an internal exchange ring and idempotents lift modulo J(R)J(R), then R   is an internal exchange ring. We also clarify some interesting results in the literature by investigating, and ultimately characterizing, the relationships between the finite (internal) exchange property, the (C3)(C3) property, and generalizations of square-free modules. We provide multiple examples delimiting these connections.